Variational and operator methods for Maxwell-Stokes system

In this paper we revisit the nonlinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions and the topology of the domain. We review and improve our recent results on existence of solutions by using the variational methods together with modified De Rham lemmas, and the operator methods. Regularity results by the reduction method are also discussed and improved.

1. Introduction. In this paper we revisit the questions of solvability and regularity of solutions to the nonlinear differential systems involving curl, including the linear, semilinear and quasilinear Maxwell system and Maxwell-Stokes system. One of the main feature of these systems is that existence of solutions depends not only on the natural of nonlinearity of the equations, but also on the type of the boundary conditions, and the topology of the domains. We shall show existence of solutions by using the variational methods, the monotone operator and compact operator method, and the reduction method. The results obtained in [48,49] will be represented, generalized and improved. This paper is organized as follows. Section 2 provides preliminary results on the operator curl , including the elementary facts about the operator curl , various spaces of vector fields, and div-curl-gradient inequalities. We shall also explain the reduction method.
In section 3 we introduce the question of effects of domain topology on the Maxwell equations and Maxwell-Stokes equations. In subsection 3.1 we show that the type of the boundary condition for the unknown potential function should be determined suitably according to the domain topology, in order for the boundary value problem to be solvable. In subsection 3.2 we briefly derive a general magnetostatic model with topology effect.
In section 5 we consider quasilinear Maxwell-Stokes systems that have variational structure. We use the minimization method together with the modified de Rham lemmas to show existence of weak solutions.
In sections 6 and 7 we study linear and semilinear Maxwell-Stokes system, and prove existence of weak solutions, then show regularity and derive estimates of the solutions.
2. Preliminaries of the operator curl .
2.1. Some facts about curl . In this subsection we briefly describe properties of the operators curl and curl 2 . The precise statements of these facts will be given in the next subsections.
1. Degenerate ellipticity of the operator curl 2 . In the three dimensions, for a vector field u = (u 1 (x), u 2 (x), u 3 (x)) T , curl 2 is a degenerately elliptic operator: if we write Hence for a boundary value problem (BVP for short) of the equations involving curl (curl equation for short), in order to have well-posedness, type of boundary conditions can not be arbitrarily prescribed. On the other hand, if Ω has no holes, curl 2 is globally elliptic in the sense that the quadratic form is coercive in the space H 1 t0 (Ω, div 0). 2. Curl and the divergence-free condition div u = 0. The operators curl , div and ∆ are connected by the equality ∆ = −curl 2 + ∇div .
If div u = 0, then curl 2 u = −∆u, which could simplify the analysis, and BVP of curl equation coupled with the divergence-free condition and with a suitable boundary condition (such as prescribing the tangential component u T ) is elliptic and satisfies the complementing condition of Agmon-Douglis-Nirenberg [1]. Unfortunately, the divergence-free condition is not always available (for instance in the Meissner model), and the requirement brings extra unknown potential term into the equation. 3. curl and domain topology. Description of the kernel and image of the operator curl is useful in the study of existence and uniqueness of BVPs of curl -equations. In a simply-connected domain a smooth vector field with its curl vanishes is a gradient of a scalar function, and in a domain without holes a smooth and divergence-free vector field is a curl of some vector field. These conclusions are not true if the domain is multiply-connected and with holes, and description of the kernel and image of curl involves the spaces H 1 (Ω) and H 2 (Ω) associated with domain topology. 4. curl and compactness. For a vector field u ∈ L 2 (Ω, R 3 ), control of both curl u and div u yields interior control of ∇u, but not globally on Ω, hence we have difficulty of lack of compactness. On a simply-connected domain without holes and with a C 2 boundary, control of both curl u and div u together with control of either normal component or tangential component of u on the boundary yields a global control of ∇u on Ω, and certain type of Poincaré inequality is available. 5. Hodge decomposition and nonlinearities. For linear systems, one often uses Hodge decomposition where φ and v satisfy certain boundary conditions according to one's needs. For nonlinear systems, Hodge decomposition does not help as much as for linear systems, but it may yield some compactness for special nonlinearities.

Spaces of functions and vector fields.
Most of materials in this subsection can be fund in [25,29]. Let us consider a domain Ω satisfying the following conditions: (O 1 ) Ω is a bounded C r domain in R 3 , r ≥ 1, and is locally situated on one side of its boundary ∂Ω, and ∂Ω has m + 1 connected components Γ j , j = 1, · · · , m + 1, where Γ m+1 is the boundary of the unbounded component of Ω c = R 3 \Ω. (O 2 ) There exist N 2-dimensional C r manifolds Σ i , non-tangential to ∂Ω, such that Σ i ∩ Σ k = ∅ for i = k, andΩ is simply-connected and Lipschitz.
The number N in (O 2 ) is called the first Betti number of Ω, which is equal to the number of handles of Ω; and the number m in (O 1 ) is called the second Betti number of Ω, which is equal to the number of holes in Ω. We say Ω is simply-connected if N = 0. We say Ω has no holes if m = 1, namely if ∂Ω is connected. We use ν to denote the unit outer normal vector of ∂Ω which points to the outside of Ω. We use C k+α (Ω), L p (Ω) and W k,p (Ω) to denote the Hölder spaces, Lebesgue spaces and Sobolev spaces for real valued functions, use C k+α (Ω, C), L p (Ω, C) and W k,p (Ω, C) to denote the corresponding spaces of complex-valued functions, and use C k+α (Ω, R 3 ), L p (Ω, R 3 ) and W k,p (Ω, R 3 ) to denote the spaces of vectors fields. However the norms both for scalar functions and for vector fields will be denoted by · C k+α (Ω) , · L p (Ω) and · W k,p (Ω) . For 1 < p < ∞, p denotes the conjugate number, and p * denotes the index of the Sobolev imbedding W 1,p (Ω) ⊂ L p * (Ω) in R 3 , namely We denote by (p * ) the conjugate number of the Sobolev index p * in R 3 , and (p ) * the Sobolev index in R 3 of the conjugate number p : (1) If F (Ω) and G(∂Ω) denote space of scalar functions on Ω or on ∂Ω, we seṫ Let ∆ denote the Laplacian operator. For 1 ≤ p ≤ ∞ we denote L p (Ω, ∆) ={u ∈ L p (Ω) : ∆u ∈ L p (Ω)}, L p (Ω, ∆0) ={u ∈ L p (Ω, ∆) : ∆u = 0}, We use u T to denote the tangential component on ∂Ω of u, namely Very often we use notation with the subscript T , for instance u 0 T , to denote a given vector field on ∂Ω which is tangential to ∂Ω. We denote H 1 (Ω) = {u ∈ L 2 (Ω, R 3 ) : curl u = 0 and div u = 0 in Ω, ν · u = 0 on ∂Ω}, H 2 (Ω) = {u ∈ L 2 (Ω, R 3 ) : curl u = 0 and div u = 0 in Ω, u T = 0 on ∂Ω}.
(5) It is well-known that dim H 1 (Ω) = N and dim H 2 (Ω) = m, where m and N are given in (O 1 ) and (O 2 ). We denote by H j (Ω) ⊥ L 2 (Ω) the orthogonal complementary of H j (Ω) in L 2 (Ω, R 3 ). When ∂Ω is of C 2 , using regularity of div-curl system we have For 1 ≤ p ≤ ∞, we define the following Sobolev type spaces of vector fields Remark 2.
Let Ω be a bounded domain in R 3 with a C 2 boundary and 1 < p < ∞.
Assume Ω is a bounded domain in R 3 with a C k+2 boundary, k is a non-negative integer, and 1 < p < ∞.
If u ∈ H 1 (Ω) ⊥ L 2 (Ω) , then the term u L p (Ω) can be dropped from the right side.
If u ∈ H 2 (Ω) ⊥ L 2 (Ω) , then the term u L p (Ω) can be dropped from the right side.
Assume Ω is a bounded domain in R 3 with a C k+2+α boundary, k is a non-negative integer, and 0 < α < 1.
The following lemma gives a description of the kernel of curl in L p (Ω, R 3 ).
Let y k = v k − ∇ψ k . Then y k ∈ H n0 (Ω, div 0) and y k → u strongly in L p (Ω, R 3 ). By the third equality in (11) we can find z k ∈ H Σ 0 (Ω, div 0) and h 1k ∈ H 1 (Ω) such that . After passing to a subsequence we may assume that h 1k → h 1 ∈ H 1 (Ω). Thus From the second line of (12) we can find such that curl w k = z k . By (16) w k is uniformly bounded in W 1,p (Ω, R 3 ). Since z k → u in L p (Ω, R 3 ), after passing to a subsequence, we may assume that w k → w weakly in W 1,p (Ω, R 3 ) and strongly in L p (Ω, R 3 ), w ∈ W 1,p t0 (Ω, div 0) and curl w = u in Ω. Therefore u ∈ curl W 1,p t0 (Ω, div 0).

Reduction method.
The reduction method is useful to study BVPs of curl systems (see [48,49]). Let us explain this method by the magneto-static model Assume H satisfies condition (H) in section 3, in particular, so curl [H(x, curl u(x)) − v(x)] = 0. From the first line in (11), there exist h 1 ∈ H 1 (Ω) and φ ∈ H 1 (Ω), such that So (22) is reduced to Taking divergence in the first equality in (24), and using the fact ν · curl u = ν · curl u T = 0, we see that φ satisfies We may get regularity and estimates of u as follows. With the v given, and taking h 1 as a parameter, we view (25) as a co-normal problem for φ and derive estimates. Then we view (24) as a div-curl system and derive estimates of u.
We may get existence of u as follows. We first fix v and h 1 , and find a solution φ = φ[v] of (25) by using, say, the monotonicity operator method. Then we find an Then So (24) with has a solution u, which is a solution of (22).
3. Maxwell equations and domain topology. The Maxwell Equations (1865) are the following where E is the electric field, B is the magnetic induction, H is the magnetic field, j is the electric current density, ρ is the charge density, ε is the permittivity. Consider (27) in R 3 , and from the second equation in (27), we can write where u is the magnetic potential. 2 Since two magnetic potentials differ by a gradient describes the same magnetic field, it has been well accepted that the field strength (E, B) (or (E, H)), instead of the electromagnetic potentials, is fundamental to describe an electromagnetic field, until Y. Aharonov and D. Bohm discovered in 1959 the observable effect of magnetic potentials [2]. As mentioned in [68] that, the work [2] raised the question of what constitutes an intrinsic and complete description of electromagnetism. Wu and Yang [68] showed that, on a multiplyconnected domain, the field strength under-describes electromagnetism and the phases (contour integrals of electromagnetic potentials) over-describes electromagnetism, but the phase factor of the integral of electromagnetic potentials provides a complete description that is neither too much nor too little.
We would like to examine the problem of complete description of electromagnetism from the PDE point of view. Since Aharonov-Bohm effect is due to nontrivial topology of the domain, our question will be the following: Question 3.1. Is it possible to find a complete description of electromagnetism by PDEs of electric field E and magnetic potential u, together with consideration of domain topology?
This question motivated our study on the following problems: (i) Examine effects of domain topology on the magneto-static problem.
(ii) Study PDEs of electric field E and magnetic potential u, with domain topology as parameters.
3.1. Boundary conditions and domain topology. We tried in [49] to understand effects of domain topology on the Maxwell system (27) by considering a quasi-static approximation. Neglecting the term ∂ ∂t (εE) in the fourth equation of (27), we get the equality curl H = j. For linear material the B-H relation is given by B = µH, where µ is the permeability. We consider nonlinear material and assume that the B-H relation is given by (12), a necessary condition for a BVP of (30) to be solvable is if Ω has no holes, if Ω has holes.
If j does not satisfy this condition, then j should be replaced by j + ∇φ such that if Ω has no holes, if Ω has holes.
Hence we consider a BVP of the Maxwell-Stokes system instead of the Maxwell system, and the boundary condition for φ should be suitably determined according to the domain topology. For a more general system, with j replaced by f (x, u), if the domain has no holes, we can formulate the problem while for a domain with holes, the BVP is where The physical meaning of the term ∇φ will be clear in Remark 3 below.
The compact operator method has been used in [48,49] to study solvability of (31) and (32). We need the following 3 if and only if w = H(x, z); and there exist positive constants µ 1 , µ 2 , C 0 such that Condition (H) is used in [49], which implies the conditions ( [49]: Let us start with (31) on a domain without holes. Given u 0 where When Ω has no holes, G is homeomorphic, and F is compact. Using the Schauder fixed point theorem we have Assume Ω is a bounded domain in R 3 without holes and with a C 3+a boundary, 0 < α < 1, H and f satisfy (H) and (f ) respectively, and u 0 and it satisfies the estimate where the constant C depends also on the constants in (H) and (f ).
If Ω has holes, we consider BVP (32). here We need the following condition on f (x, u): , and the following holds uniformly for x ∈Ω: This condition implies conditions (f 1 ) and (f 2 ) in [49].

3.2.
A magneto-static model with domain topological effect. A general magneto-static model which includes the effect of domain topology was derived in [51] under the following assumptions: (i) The vector-valued function H(x, ·) in the H-B relation (29) has an inverse B(x, ·). (ii) The current density j satisfies the Ohm's law 4 where σ ≥ 0 is the electric conductivity, and j a is the applied current density. We first derive a quasi-static approximation of the Maxwell system (27). Assume that From (27) div B = 0. By the second line of (11) we write where w(t, ·) ∈ H 1 n0 (Ω, div 0), h 2 (t, ·) ∈ H 2 (Ω). Then from the third line of (27) we have So ∂ t h 2 = 0 and curl (E + ∂ t w) = 0. By the first line of (11) Using the fourth line of (27), (29) and (36), we have Set in Ω for all t ≥ 0. Neglecting the displacement current ∂ t (εE) in (37), we get the following Problem. Quasilinear parabolic Maxwell-Stokes system Now we derive time-independent model. Assume u,h 1 , j a , q, ε and ρ are independent of t, σ is a positive constant, and denote We have Problem. Quasilinear elliptic Maxwell-Stokes system Remark 3. About the physical meaning of ∇φ in (40): Mathematically, the gradient term ∇φ is necessarily introduced into the equation in (40) to balance the equality, which is necessary for solvability of the BVP. On the other hand, from so ∇φ represents both electric field E and domain topology through h 1 .
Now we consider solvability of (40), with boundary condition φ = 0 on ∂Ω, and u satisfies either the tangential trace boundary condition or the tangential curl boundary condition Assume j a ∈ H 2 (Ω) ⊥ L 2 (Ω) , which is a necessary condition for (40) to have a solution (see [48,Lemma 2.1]). Then by the first equality in (14) we write In the following we assume that Ω is a bounded domain in R 3 with a C 2 boundary, H satisfies condition (H), First we consider problem (43)- (41).
Proof. Since J + h 1 ∈ H Γ (Ω, div 0), there exists a unique (43)-(41) is solvable if and only if there exist φ ∈ H 1 (Ω), h 1 ∈ H 1 (Ω), such that the following has a solution v: From the second equality in (12) we know that, for the given j 0 and h 1 , (45) is solvable if and only if the following co-normal problem has a solution φ which satisfies the orthogonality condition Choose an orthonormal basis By the monotone operator method we can show that, (46) has a unique solution Then (47) can be written as f (ξ) = c. As in [48, proof of Theorem 4.9], we can show that f (ξ) is a continuous map in R N , and for large R it holds that By the accurate angle theorem for continuous maps in Let φ ξ0 be the solution of (46) associated with h 1,ξ0 . Then (h 1,ξ0 , φ ξ0 ) solves problem (46)-(47).

Proposition 4 ([51]). Assume
. By (15), both w H 1 (Ω) and curl w L 2 (Ω) are equivalent norms in X. By Lax-Milgram lemma, there is a continuous and strongly monotone operator A h2 : X → X * such that By Browder-Minty theorem, A h2 is surjective, so there exists a unique u ∈ X such that For any w ∈ H 1 (Ω, R 3 ) we can write

Ω). Since
A h2 (u), h + ∇φ X * ,X = 0, and using the last equality in (48), we see that (49) holds also for h + ∇φ, hence it holds for w. (Ω, div 0) can be represented by a gradient ∇φ with the potential φ ∈ L 2 (Ω). Note that no information on trace of φ on ∂Ω is given in the lemma. 5 A modified de Rham's lemma for functionals vanishing on H 1 t0 (Ω, div 0) was given in [49], where the trace on ∂Ω of the potential φ is identified. As mentioned in [49, p.10] that, the statement of the lemma and its proof given there remain valid for functionals vanishing in W 1,p t0 (Ω, div 0) for 1 < p < ∞ Here, we give a precise statement and complete proof of them.
Let Ω be a bounded domain in R 3 with a C 2 boundary and 1 < p < +∞.
Then ψ ∈L p (Ω), Since Ω ψdx = 0, we can find v ψ ∈ W 1,p 0 (Ω, Then , from which and since p − p /p = 1, we get Combining (i), (53) and (54) we have For φ ∈ L p (Ω) we can also define a functional T φ on W 1,p t0 (Ω, R 3 ) by In the same manner we define T φ,c for φ ∈ L p (Ω) and c ∈ R (see (50)). If furthermore φ has zero trace on ∂Ω then we may write T φ as a "gradient", T φ = ∇φ, which is understood as a functional on W 1,p t0 (Ω, R 3 ) . If φ does not have zero trace, when considered as a functional on W 1,p 0 (Ω, R 3 ), one may write T φ as gradient ∇φ, However, when considered as a functional on W 1,p t0 (Ω, R 3 ), writing T φ as gradient ∇φ may cause some confusion. Let Ω be a bounded domain in R 3 with a C 2 boundary and 1 < p < +∞. Assume that T ∈ W 1,p, *
Step 1. We first show that if T ∈ W 1,p, *

t0
(Ω, R 3 ) is such that (55) holds, then there exist φ ∈L p (Ω) and ζ ∈ W −1/p ,p (∂Ω) such that T = T φ,ζ . The proof is a modification of the proof for the classical de Rham lemma (see for instance [13, p.243]). If X is a Banach space and X * is its dual space, for A ⊂ X we denote (Ω).
We need prove Z ⊥ ⊆ T. Since X is reflexive, we know that and we only need to show that T ⊥ ⊆ Z. Let u ∈ T ⊥ . From (55), for any ψ ∈ L p (Ω) and ξ ∈ W −1/p ,p (∂Ω) we have Taking ψ ∈ W 1,p 0 (Ω) and ξ = 0 in the above equality we see that div u = 0 in the weak sense, hence u ∈ W 1,p t0 (Ω, div 0) = Z. This verifies that T ⊥ ⊆ Z.

A modified de Rham lemma in
Let Ω be a bounded domain in R 3 with a C 2 boundary and 1 < p < ∞.
Proof. The proof is similar to the proof of Lemma 4.2 but simpler. Let It implies that div u = 0, hence u ∈ W 1,p n0 (Ω, div 0) = Z. Thus T ⊥ ⊆ Z, hence Z ⊥ ⊆ T.
For simplicity of notation we shall write T curl w by curl w, so we have curl w, v W 1,p, *
As an example assume A satisfies (A 0 ), u ∈ W p (Ω, R 3 ) and g ∈ L p (Ω, R 3 ). Then w = A curl u − g ∈ L p (Ω, R 3 ), and we have In particular we have In the following we assume that Let Ω be a bounded domain in R 3 with a Lipschitz boundary, p, r, A curl u, g, U satisfy (61), such that Ω { A curl u + g, curl z + U, z }dx = 0 ∀z ∈ W 1,p 0 (Ω, div 0).

4.5.
Representation of a curl -functional vanishing on W 1,p n0 (Ω, div 0). We need the following facts. Let Ω be a bounded domain in R 3 with a C 2 boundary and 1 < p < ∞.
Step 2. For any w ∈ W 1,p (Ω, R 3 ), let ψ w ∈ W 2,p (Ω) be such that Taking v = w − ∇ψ w ∈ W 1,p n0 (Ω, R 3 ) as a test field in (74), and using the facts Although ψ w is not uniquely determined by w, for any two functions ψ w ∈ W 2,p (Ω) satisfying condition (75) for the same w, we have ψ So the integral Ω h∆ψ w dx depends only on ν · w.
The above lemma (when p = 2) makes it possible to solve the following BVP by minimizing the associated energy functional on H 1 n0 (Ω, div 0).

Remark 5.
In Lemma 4.7 the function φ is unique up to an additive constant. It suggests that in (78) we should not pose any extra boundary condition on φ.

Example. Consider equations (81) with
where h(t) is a continuous positive function on (0, +∞). We can write H(z) = ∇P (z), where This example includes the p-curl curl system with q = p − 2 and h(s) = s.
We shall also consider Maxwell-Stokes problem with the natural boundary condition where 1 < p < +∞. In (83) no boundary condition on φ is prescribed, see Remark 5.

Existence by minimization.
To avoid technical complication, we assume the following conditions: z) is strictly convex in z, and there exist positive constants C 1 , C 2 , such that , and there exist 1 < p 1 < p * and a positive constant C 3 such that (c) c ∈ C(Ω), c(x) ≥ 0 on Ω, and if Ω has holes then assume Assume Ω is a bounded domain in R 3 with a C 2 boundary, 1 < p < ∞, P , F , c and u 0 T satisfy (P p ), (F p ), (c) and (U 0 ) respectively. Then (79) has a weak solution (u, φ) ∈ W 1,p t (Ω, div 0, u 0 T ) × W 1,r 0 (Ω), where r = p * /(p 1 − 1).
Case 2. Ω has holes and c(x) ≥ c 0 > 0 on Ω. Take 0 < ε < c 0 /(2p) We have So curl u j L p (Ω) + u j L p (Ω) is bounded. Using (16) we see that {u j } is bounded in W 1,p (Ω, R 3 ). The rest of the proof is same as in case 1.
(i) When p = 2, the weak solutions of the quasilinear Maxwell-Stokes system (79) have Schauder regularity if the data are suitably smooth (see [49]): For the Maxwell type equations, Schauder regularity of weak solutions has been established for many quasilinear models, see [10,46,47,35,50] for the model of Meissner states of superconductivity, [48,49] for the nonlinear Maxwell equations.
Proof. Extend a(x, t) and g(x, t) evenly for t < 0. Set Then From (a) and (g) we see that P (x, z) and F (x, u) satisfy (P p ) and (F p ) with p 1 = p, and for ξ = 0, So P (x, z) is strictly convex in z, and existence follows from Theorem 5.3.
If a Maxwell-Stokes system has a weak solution (u, φ), and if we can use the special structure of the equation to show that ∇φ = 0, then u is a solution of the corresponding Maxwell system. As an example, we examine solvability of the following Maxwell system in Ω, where J is given.
Proof. By Theorem 5.3 the following problem has a weak solution (u, φ) with φ ∈ W 1,r 0 (Ω): If either (i) or (ii) holds, from the equation we see that ∆φ = 0 in the sense of distribution. Since φ = 0 on ∂Ω, we see that φ ≡ 0 on Ω. Hence u is a weak solution of (90).
6. Linear Maxwell-Stokes system. We review and generalize some existence and regularity results in [49,Appendix B] for the linear Maxwell-Stokes system curl (A curl u) = J + ∇φ, div u = 0 in Ω, and the linear Maxwell type system curl (A curl u) = J, div u = 0 in Ω, Note that in the case where the domain Ω has holes, then the operator u → curl (A curl u) is degenerate, which causes difficulties in analysis. One could add a positive term Bu to the left side of the equation in (91) and consider the equation of the form curl (A curl u) + Bu = J + ∇φ, where B ∈ C 0 (Ω, S + (3)), then the corresponding BVP is non-degenerate and existence problem becomes much easier. However in this section we are interested in the degenerate case, and we wish to examine new phenomena for (91) and (92). BVPs of the type (92) have been studied by many authors, see the classical works [25,16,40] and the references therein. For the recent research works, see for instance [54,38,4,71,7,6], just name a few.
Proof. The case where p = 2 has been proved in [49,Appendix B]. Here we consider the general case where 1 < p < ∞.
By conditions (A 0 ) and (B 0 ), the following eigenvalue problem has a sequence of eigenvalues {λ n } ∞ n=1 with 0 < λ 1 ≤ λ 2 ≤ · · · , each has finite multiplicity, and λ n → ∞. So when F (x, u) is even in u, we can use the method in the proof of [60, p.112, Theorem II.6.6] and use Lemma 4.6 to get the conclusion.
Among semilinear Maxwell type systems, the model of Meissner states of superconductivity is an interesting system −λ 2 curl 2 A = (1 − |A| 2 )A in Ω, where Ω is a bounded domain in R 3 . Note that (129) does not have the divergencefree requirement. In fact the solution A of (129) is unique if H e T is small [10], so we can not require div A = 0. On the other hand, for small solutions (129) can be transferred to a system with divergence-free condition. In fact, if A is a solution of (129) and where F (u) is a positive valued function defined on [0, 2/ √ 27]. When Ω is simplyconnected and without holes, existence of the solution to (130) with small boundary data is proved in [41]. Existence and regularity of the solution for boundary data under the optimal bound H e T ∈ C 2+α (∂Ω, R 3 ), H e T C 0 (∂Ω) <

18 ,
is obtained in [10], and the concentration behavior of the solution as λ → 0 is also established. The restriction for the domain to be simply-connected and without holes is removed in [35], hence the results in [10] remain true for a general bounded domain. More precise results of the location of concentration points is proved in [70], and the full model of Meissner states are studies in [44,50]. These results for (130) give the corresponding results for (129). However (129) and (130) are not equivalent for large solutions with A L ∞ (Ω) > 1/ √ 3.
Regarding question (i), existence of solutions on radially symmetric domains has been obtained in [69], while the problem for a general domain remains open. Question (ii) is related to the vortex solutions of Ginzburg-Landau equations for superconductivity [17], and we guess that there may exist solutions which satisfy max|A(x)| = 1 and have singularities. This could happens in the 2-dimensional problem. In fact, the vector field in [52, p.5037, (1.12)] satisfies |v(x)| ≡ 1 and curl v = 2 sin φ 0 , a constant, hence v is a solution of the corresponding 2-dimensional problem on the disc, but v does not lie in H 1 (Ω, R 2 ).
It is interesting to compare (129) with the following equation was proposed in the study of the asymptotic limit of nematic liquid crystals [52, p.5037] and [43,p.380,Problem 4], and very little is known.
Remark 7. In this paper we require the domain to have a C 2 or C 2+α boundary. It will be interesting to study these nonlinear problems on Lipschitz domains, which is interesting both in mathematics and in engineering.